Notebook

Buckingham $\pi$: Dimensional Analysis¶

Mokbel Karam, and Prof. Tony Saad (www.tsaad.net)
Department of Chemical Engineering
University of Utah

In [1]:

from IPython.display import Image
from IPython.core.display import HTML 

In this Jupyter notebook we will execute the code presented in the paper.

Example 1: Pressure Inside a Bubble¶

Using mass, length and time as fundamental physical dimensions:

In [2]:

from buckinghampy import BuckinghamPi

Pressure_In_Bubble = BuckinghamPi()
Pressure_In_Bubble.add_variable(name='{\\Delta}p',units='M*L^(-1)*T^(-2)')  # pressure 
Pressure_In_Bubble.add_variable(name='R',units='L')                         # diameter
Pressure_In_Bubble.add_variable(name='\\sigma',units='M*T^(-2)')            # surface tension
try:
    Pressure_In_Bubble.generate_pi_terms()
    Pressure_In_Bubble.print_all()
except Exception as e:
    print(e)

The number of variables has to be greater than the number of physical dimensions.

Using force and length as fundamental physical dimensions:

In [3]:

from buckinghampy import BuckinghamPi

Pressure_In_Bubble = BuckinghamPi()
Pressure_In_Bubble.add_variable(name='{\\Delta}p',units='F*L^(-2)')       # pressure 
Pressure_In_Bubble.add_variable(name='R',units='L')                       # diameter
Pressure_In_Bubble.add_variable(name='\\sigma',units='F*L^(-1)')          # surface tension

Pressure_In_Bubble.generate_pi_terms()
Pressure_In_Bubble.print_all()

  sets  Pi 1
------  --------------------------
     1  \frac{\sigma}{R {\Delta}p}

Example 2: Pressure Drop in a Pipe¶

In [4]:

from buckinghampy import BuckinghamPi

Pressure_Drop = BuckinghamPi()
Pressure_Drop.add_variable(name='{\\Delta}p',units='M*L^(-1)*T^(-2)')     # pressure drop
Pressure_Drop.add_variable(name='R',units='L')                            # length of the pipe
Pressure_Drop.add_variable(name='d',units='L')                            # diameter of the pipe
Pressure_Drop.add_variable(name='\\mu',units='M*L^(-1)*T^(-1)')           # viscosity
Pressure_Drop.add_variable(name='Q',units='L^(3)*T^(-1)')                 # volumetic flow rate

Pressure_Drop.generate_pi_terms()
Pressure_Drop.print_all()

  sets  Pi 1                                                     Pi 2
------  -------------------------------------------------------  -------------------------------------------------------
     1  \frac{R}{d}                                              \frac{Q \mu}{d^{3} {\Delta}p}
     2  \frac{R \sqrt[3]{{\Delta}p}}{\sqrt[3]{Q} \sqrt[3]{\mu}}  \frac{d \sqrt[3]{{\Delta}p}}{\sqrt[3]{Q} \sqrt[3]{\mu}}
     3  \frac{d}{R}                                              \frac{Q \mu}{R^{3} {\Delta}p}

Example 3: Economic Growth¶

In [5]:

from buckinghampy import BuckinghamPi

Economic_Growth = BuckinghamPi()
Economic_Growth.add_variable(name='P',units='K',explicit=True)           # capital
Economic_Growth.add_variable(name='L',units='Q/T')                       # labor per period of time
Economic_Growth.add_variable(name='{\\omega_{L}}',units='K/Q')           # wages per labor
Economic_Growth.add_variable(name='Y',units='K/T')                       # profit per period of time
Economic_Growth.add_variable(name='r',units='1/T')                       # rental rate period of time
Economic_Growth.add_variable(name='{\\delta}',units='1/T')               # depreciation rate

Economic_Growth.generate_pi_terms()
Economic_Growth.print_all()

  sets  Pi 1                               Pi 2                      Pi 3
------  ---------------------------------  ------------------------  ------------------
     1  \frac{P r}{L {\omega_{L}}}         \frac{Y}{L {\omega_{L}}}  \frac{{\delta}}{r}
     2  \frac{P {\delta}}{L {\omega_{L}}}  \frac{Y}{L {\omega_{L}}}  \frac{r}{{\delta}}
     3  \frac{P r}{Y}                      \frac{L {\omega_{L}}}{Y}  \frac{{\delta}}{r}
     4  \frac{P {\delta}}{Y}               \frac{L {\omega_{L}}}{Y}  \frac{r}{{\delta}}